## The need for play

I have just finished reading Teaching Mathematics: Toward a Sound Alternative by Brent Davis.

In this transformative book, Davis describes the need for (Mathematics) teachers to

**allow space for play**in their classrooms, reminding me of Roger Penrose's view of Mathematics as playing.

Davis chooses not to give a clear definition for the word play in this context, instead choosing to maintain the ambiguity presented by its many uses. He describes the essence of play as a movement back and forth, related to the "

*as yet unexperienced, the unpredictable or uncontrollable*".

Davis talks of the need to allow space to play in order to create and embrace possibilities:

Amid the possibilities that present themselves, the teaching here becomes a matter of discerning and selecting what seems important... [The teacher] is not a teller, not a facilitator, but a player, a participant.

The teacher, in allowing a space for play, opens up a mathematically rich space. She is, at this point, able to move in a number of directions...

Listening - that attitude of open-ness to the possibilities that continuously present themselves - is essential to teaching in this conception.

The key idea for teaching here is, "

*not the making of progress, but the allowing of space for movement,*" of which negotiation, conversation and listening are fundamental parts.

## Why do we like to ... play? ... do Mathematics?

While reading Davis's book, I decided to ask @MissJones_14 why she likes playing certain board games. Here are the reasons she gave:

- Understanding complex sets of rules, trying to work out how the game works.
- Learning the (often unusual and slightly ridiculous) language of the game, and having conversations with others using this language.
- Identifying strategies as you go along, trying them out, working out if they are going to work or whether they need to be adapted - this is only fun if the game is complex enough.
- Watching others' strategies, seeing how they work and whether they are successful.
- Sensory aspects of the game, e.g. pieces and tiles that are pleasant to touch and move, that are aesthetically pleasing.
- Enjoyment in winning, especially if the game is unpredictable and new to those playing. The enjoyment lessens as the game, and strategies, become 'known'.

Interestingly, I could have given any of these reasons had I been asked "Why do I like doing Mathematics?" (although I would replace winning with solving a problem).

The question is then - how can I create the conditions for play in my teaching?

## Creating tasks with the space for play

A large part of allowing the space for play is through listening to students, creating a culture of conversation in the classroom. However, another large part of this approach is the type of tasks that are used; they must be designed to allow a range of possibilities.

Here are some tasks I have designed for use this week that I hope will create a range of possible student responses, from which we can start conversations about the mathematics underlying the equations of straight lines.

The main objective here is to talk about equations of parallel and perpendicular lines.

We have not talked a great deal about what constitutes a proof. I will be interested to see what the students come up with for their proofs, and what we are convinced by. I am anticipating that they will attempt to prove that (say) AB and AD are perpendicular using a square-counting/gradient approach. Following the task, we will look at the equations of the perpendicular and parallel lines.

The second part is interesting because they may use triangle areas, or perhaps Pythagoras' Theorem to find the area. Or perhaps they will try something completely different.

The third part is interesting because I think it is not possible for a rectangle with 'integer vertices' (we might try and prove this). I am interested in how they go about attempting this - will they try and draw lots of rectangles, or something else? Will they realise it is not possible with integer vertices, and draw vertices on non-integer co-ordinates?

The objective is to be comfortable with using the ax + by + c = 0 form of straight line equations.

I have chosen the lines to be intentionally 'fractional' as I want them to practice operations on fractions, something they find difficult. They will do this in pairs, and we will have a discussion about how we know which is which. I anticipate some students rearranging the equations into the form y = mx + c, but perhaps some might substitute x = 0 or y = 0 and compare axis-intercepts?

I have left the end of the task open to see what they consider important. I anticipate they might try to find intersections with axes (I would us to notice that putting x = 0 and y = 0 into this form of the equation leads us to a simple calculation of the axis-intersections), and perhaps intersections between lines. Others might choose to find gradients.

I am hoping that both these tasks with allow students the space to play, and will promote conversation and possibilities, as well as providing practice on the mathematical content and give opportunities to improve problem solving ability.

I have chosen the lines to be intentionally 'fractional' as I want them to practice operations on fractions, something they find difficult. They will do this in pairs, and we will have a discussion about how we know which is which. I anticipate some students rearranging the equations into the form y = mx + c, but perhaps some might substitute x = 0 or y = 0 and compare axis-intercepts?

I have left the end of the task open to see what they consider important. I anticipate they might try to find intersections with axes (I would us to notice that putting x = 0 and y = 0 into this form of the equation leads us to a simple calculation of the axis-intersections), and perhaps intersections between lines. Others might choose to find gradients.

I am hoping that both these tasks with allow students the space to play, and will promote conversation and possibilities, as well as providing practice on the mathematical content and give opportunities to improve problem solving ability.